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Trigonometry : Triangles

Study concepts, example questions & explanations for Trigonometry

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6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Finding Angles

What angle does the ramp make with the bottom of the stair?

Screen_shot_2014-02-15_at_3.16.35_pm

Possible Answers:

56.3^o

33.7^o

$\small 28.9^o$

30^o

45.0^o

Correct answer:

33.7^o

Explanation:

$\small tan(c)=\frac{c}{a}=\frac{2}{3}$

$\small tan(c)=\frac{2}{3}$

$\small c=tan^{-1}(\frac{2}{3})$

c=33.7^o

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Example Question #3 : Finding Angles

Given that angle A is 23.0^o, side a is 1.43 in., and side b is 3.62 in., what is the angle of B?

Screen shot 2014 02 15 at 3.41.09 pm

Possible Answers:

83.6^o

70^o

90^o

81.5^o

55^o

Correct answer:

81.5^o

Explanation:

$\small \small \small \frac{sin(23.0^o)}{1.43\: in.}=\frac{sin(B)}{3.62\: in.}$

$\small \small \small \sin(B)=\frac{3.62 in.\times \sin(23.0^o)}{1.43 in.}$

$\small \small \small \sin(B)=0.99$

$\small B=sin^{-1}(0.99)$

$\small B=81.5^o$

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Example Question #367 : Trigonometry

Artemis wants to build a ramp to make the entrance to their home more accessible. The angle between the ramp and the ground cannot be more than $7^\circ$ steep. Artemis has feet of space in their yard that the ramp can take up, and the distance between the ground and the house entrance is feet high. Will Artemis be able to build a ramp that complies with the $\leq7^\circ$ standard?

Possible Answers:

Yes

Correct answer:

Explanation:

Begin the problem by visualizing a diagram of the situation:

Screen shot 2020 08 09 at 3.36.03 pm

We can use inverse trig to solve for the unknown angle $\theta$ .

$\tan\theta=\frac{3}{12}$

$\theta=\tan^-^1\left (\frac{3}{12} \right )$

$\theta=14.04^\circ$

Because this angle is larger than $7^\circ$ , this ramp would not comply with standards.

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Example Question #368 : Trigonometry

If the hypotenuse of a right triangle has a length of 42.29 meters, and the length of a leg is 12.88 meters, what is the angle between the hypotenuse and the leg?

Possible Answers:

$15.15^\circ$

$14.64^\circ$

$74.85^\circ$

$99.99^\circ$

Correct answer:

$74.85^\circ$

Explanation:

The leg must be an adjacent side to the hypotenuse.

Therefore, we can use inverse cosine to solve for the angle.

First write the equation for sine of an angle.

$cos(\theta)=\frac{\textup{Adjacent}}{\textup{Hypotenuse}}$

$\theta=cos^{-1}\left(\frac{\textup{Adjacent}}{\textup{Hypotenuse}}\right)$

Substitute the lengths given and solve for the angle.

$\theta=cos^-^1\left (\frac{12.88}{49.29} \right )=74.85^\circ$

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Example Question #111 : Triangles

If the height of the stair is 2 ft, and the length of the stair is 3 ft, how long must the ramp be to cover the stair?

Screen_shot_2014-02-15_at_2.24.54_pm

Possible Answers:

3.6 ft

13 ft

2.5 ft

5 ft

10 ft

Correct answer:

3.6 ft

Explanation:

Use the Pythagorean triangle to solve for the third side of the triangle.

a^2+b^2=c^2

2^2+3^2=13

$\therefore c^2=13$

$c=\sqrt{13}$

Simplify and you have the answer:

c=3.6 ft

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Example Question #2 : Finding Sides

A surveyor looks up to the top of a mountain at an angle of 35 degrees. If the surveyor is 2400 feet from the base of the mountain, how tall is the mountain (to the nearest tenth of a foot)?

Possible Answers:

1680.5 feet

3427.6 feet

2135.8 feet

1966.0 feet

1376.6 feet

Correct answer:

1680.5 feet

Explanation:

This is a typical angle of elevation problem.

From the question, we can infer that the ratio of the mountain height to the surveyor's horizontal distance from the mountain is equal to the tangent of 35 degrees - the mountain height is the "opposite side", while the horizontal distance is the "adjacent side". In other words:

$tan (35^{\circ}) = \frac{mountain\ height}{ 2400\ ft}$

Solving for the mountain height gives a vertical distance of 1680.5 feet.

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Example Question #112 : Triangles

In a right triangle, the legs are and . What is the length of the hypotenuse?

Possible Answers:

500

Correct answer:

Explanation:

Here, we can use the Pythagorean Theorem.

This states that the sum of the squares of the legs is equal to the square of the hypotenuse.

Thus, we plug into the formula to get our hypotenuse.

$7^{^{2}}+24^{2} = c^{2}$

$c=\sqrt{49+576}=\sqrt{625}=25$

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Example Question #1 : Finding Sides

If the altitude of a triangle has a height of 5, and also bisects a 60 degree angle, what must be the perimeter of the full triangle?

Possible Answers:

$2\sqrt3$

$5\sqrt3$

$10\sqrt3$

$\frac{5\sqrt3}{3}$

$\frac{ 10\sqrt3}{3}$

Correct answer:

$10\sqrt3$

Explanation:

The altitude of the triangle splits the 60 degree angle into two 30 degree angles. Notice that this will create two right triangles with 30-60-90 angles. This indicates that the full triangle is an equilateral triangle.

The altitude represents the adjacent side of the split triangles. The bisected angle is 30 degrees. Find the hypotenuse of either split triangle. This is the side length of the equilateral triangle.

$cos(\theta)=\frac{A}{H}$

$H=\frac{5}{cos(30)}= \frac{5}{\frac{\sqrt3}{2}}= \frac{5 \times 2}{\sqrt{3}} = \frac{10}{\sqrt3}$

Rationalize the denominator.

$\frac{10}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{ 10\sqrt3}{3}$

Since the full triangle has three sides, multiply this value by three for the perimeter.

$3\left(\frac{ 10\sqrt3}{3}\right)=10\sqrt3$

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Example Question #3 : Finding Sides

A horizontal ramp of unknown length meets the ground at angle of 20°. The height of the ramp is 6 feet and makes a right angle with the ground. What is the length of the ramp?

Possible Answers:

$22.4\:\uptext{ft}$

$15.5\:\uptext{ft}$

$25.5\:\uptext{ft}$

$20.5\:\uptext{ft}$

None of the other answers.

Correct answer:

None of the other answers.

Explanation:

The question is looking for the length of the ramp which would be the length of the part one would walk on and is also synonymous with the hypotenuse of our right triangle. With the given angle and opposite side we can use the sine function to solve for the length of the ramp because

$\sin\theta =\frac{opp}{hyp}$ .

We have all but the length of the hypotenuse.

Solve for the hypotenuse:

$hyp=\frac{opp}{\sin\theta }$

Plug in the angles and sides given:

$hyp=\frac{6}{\sin20 }=17.5\:\uptext{ft}$

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Example Question #3 : Finding Sides

A zipline is attached from the edge of the roof of a 23 foot high building and meets the ground 23 feet from the base of the building. Find the angle $\theta$ of elevation the cable makes with the ground.

Possible Answers:

$\theta=35^{\circ}$

$\theta=45^{\circ}$

$\theta=49^{\circ}$

$\theta=25^{\circ}$

$\theta=34^{\circ}$

Correct answer:

$\theta=45^{\circ}$

Explanation:

Concerning the angle of elevation which we are to find, we know the side length opposite this angle and the side length adjacent to it.

We use

$\tan=\frac{opp}{adj}$ .

$\tan\theta=\frac{opp}{adj}=\frac{23}{23}=1$

Take the inverse of tan to find the corresponding angle:

$\tan^{-1}(1)=45^{\circ}$

A short cut for solving is to realize right triangles with equal sides must have two 45° angles!

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Trigonometry : Triangles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #2 : Finding Angles

Example Question #3 : Finding Angles

Example Question #367 : Trigonometry

Example Question #368 : Trigonometry

Example Question #111 : Triangles

Example Question #2 : Finding Sides

Example Question #112 : Triangles

Example Question #1 : Finding Sides

Example Question #3 : Finding Sides

Example Question #3 : Finding Sides

All Trigonometry Resources

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